5x + 7y = 1
ax + by = 1
In the given pair of equations, a and b are constants. The graph of this pair of equations in the xy-plane is a pair of perpendicular lines. Which of the following pairs of equations also represents a pair of perpendicular lines?

Question

5x + 7y = 1
ax + by = 1

In the given pair of equations, a and b are constants. The graph of this pair of equations in the xy-plane is a pair of perpendicular lines. Which of the following pairs of equations also represents a pair of perpendicular lines?

Accepted Answer

Choice B is correct. Two lines are perpendicular if their slopes are negative reciprocals, meaning that the slope of the first line is equal to −1 divided by the slope of the second line. Each equation in the given pair of equations can be written in slope-intercept form, y = mx + b, where m is the slope of the graph of the equation in the xy-plane and (0,b) is the y-intercept. For the first equation, 5x + 7y = 1, subtracting 5x from both sides gives 7y = −5x + 1, and dividing both sides of this equation by 7 gives y = − 5 ⁄ 7 x + 1 ⁄ 7 . Therefore, the slope of the graph of this equation is − 5 ⁄ 7 . For the second equation, ax + by = 1, subtracting ax from both sides gives by = −ax + 1, and dividing both sides of this equation by b gives y = − a ⁄ b x + 1 ⁄ b . Therefore, the slope of the graph of this equation is − a ⁄ b . Since the graph of the given pair of equations is a pair of perpendicular lines, the slope of the graph of the second equation, − a ⁄ b , must be the negative reciprocal of the slope of the graph of the first equation, − 5 ⁄ 7 . The negative reciprocal of − 5 ⁄ 7 is 7 ⁄ 5 . Therefore, − a ⁄ b = 7 ⁄ 5 , or a ⁄ b = − 7 ⁄ 5 . Similarly, rewriting the equations in choice B in slope-intercept form yields y = − 10 ⁄ 7 x + 1 ⁄ 7 and y = − a ⁄ 2b x + 1 ⁄ 2b . It follows that the slope of the graph of the first equation in choice B is − 10 ⁄ 7 and the slope of the graph of the second equation in choice B is − a ⁄ 2b . Since a ⁄ b = − 7 ⁄ 5 , − a ⁄ 2b is equal to −( 1 ⁄ 2 )(− 7 ⁄ 5 ), or 7 ⁄ 10 . Since 7 ⁄ 10 is the negative reciprocal of − 10 ⁄ 7 , the pair of equations in choice B represents a pair of perpendicular lines. Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

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